论文标题
某些随机矩阵合奏的结构函数的差分身份
Differential identities for the structure function of some random matrix ensembles
论文作者
论文摘要
随机矩阵集合的结构函数可以指定为线性统计$ \ sum_ {j = 1}^n e^{ik_1λ_j} $，$ \ sum_ {特征矩阵$θ_j$用于单一矩阵。因此，它可以用密度密度相关的傅立叶变换$ρ_ {（2）} $写成。对于单一矩阵的圆形$β$安装，我们将$ρ_{（2）} $的散装缩放限制为$β+ 1 $的线性微分方程的解决方案 - duality duality with $ρ_{（2）} $替换为$β$ $β$ 4/β。从此特征中获得的$β= 6 $在此情况下与先前确定的结果相结合，以确定$β/2 $中的10 palindromic多项式的明确形式，该形式确定了$ | k | |^{11} $在小$ | k |在一般$ bentery $ ebemep $ beply $ ebe的结构中的$ | K |^{11} $。对于高斯统一的合奏，由于俄克拉山脉，我们对近期的推导和概括进行了重新研究，该身份将结构函数与Brézin和hikami最初以随机基质理论得出的Laguerre单一合奏中的简单量相关。这用于确定各种缩放限制，其中许多与浸入式 - 斜纹植物的效应有关，在最近对许多身体量子混乱的研究中强调了倾斜度，并且也允许建立收敛速率。
The structure function of a random matrix ensemble can be specified as the covariance of the linear statistics $\sum_{j=1}^N e^{i k_1 λ_j}$, $\sum_{j=1}^N e^{-i k_2 λ_j}$ for Hermitian matrices, and the same with the eigenvalues $λ_j$ replaced by the eigenangles $θ_j$ for unitary matrices. As such it can be written in terms of the Fourier transform of the density-density correlation $ρ_{(2)}$. For the circular $β$-ensemble of unitary matrices, and with $β$ even, we characterise the bulk scaling limit of $ρ_{(2)}$ as the solution of a linear differential equation of order $β+ 1$ -- a duality relates $ρ_{(2)}$ with $β$ replaced by $4/β$ to the same equation. Asymptotics obtained in the case $β= 6$ from this characterisation are combined with previously established results to determine the explicit form of the degree 10 palindromic polynomial in $β/2$ which determines the coefficient of $|k|^{11}$ in the small $|k|$ expansion of the structure function for general $β> 0$. For the Gaussian unitary ensemble we give a reworking of a recent derivation and generalisation, due to Okuyama, of an identity relating the structure function to simpler quantities in the Laguerre unitary ensemble first derived in random matrix theory by Brézin and Hikami. This is used to determine various scaling limits, many of which relate to the dip-ramp-plateau effect emphasised in recent studies of many body quantum chaos, and allows too for rates of convergence to be established.
